行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}1 & {-}1 & 2 & 0 \\ {-}1 & 1 & 2 & 1 \\ {-}2 & 0 & 1 & 1 \\ 2 & 0 & 1 & {-}1\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & 2 & 0 \\ {-}1 & 1 & 2 & 1 \\ {-}2 & 0 & 1 & 1 \\ 2 & 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [0, 1, 1, -1]

\( \qquad\quad = (-1)^{4-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}2 & 0 & 1 \\ 2 & 0 & 1\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}1 & 1 & 2 \\ 2 & 0 & 1\end{array}\,\right| + (-1)^{4-4}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}1 & 1 & 2 \\ {-}2 & 0 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-2}\times1\times(-4) + (-1)^{4-3}\times1\times(-8) + (-1)^{4-4}\times(-1)\times8 \)

\( \qquad\quad = -4 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}2 & 0 & 1 \\ 2 & 0 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 1, 1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & 0\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times1\times2 + (-1)^{3-3}\times1\times(-2) \)

\( \qquad\quad = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & 0\end{array}\,\right| = (-2)\times0 - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & 0\end{array}\,\right| = 1\times0 - (-1)\times(-2) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}1 & 1 & 2 \\ 2 & 0 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, 1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times(-2) + (-1)^{3-2}\times2\times2 + (-1)^{3-3}\times1\times0 \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - 1\times2 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & 1\end{array}\,\right| = 1\times1 - (-1)\times(-1) = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & 2 \\ {-}1 & 1 & 2 \\ {-}2 & 0 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, 1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ {-}2 & 0\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & 0\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times2 + (-1)^{3-2}\times2\times(-2) + (-1)^{3-3}\times1\times0 \)

\( \qquad\quad = 8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ {-}2 & 0\end{array}\,\right| = (-1)\times0 - 1\times(-2) = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & 0\end{array}\,\right| = 1\times0 - (-1)\times(-2) = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & 1\end{array}\,\right| = 1\times1 - (-1)\times(-1) = 0 \)